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Properties

Label 6025.2.a.i
Level $6025$
Weight $2$
Character orbit 6025.a
Self dual yes
Analytic conductor $48.110$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6025,2,Mod(1,6025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6025.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2 x^{14} - 16 x^{13} + 31 x^{12} + 99 x^{11} - 184 x^{10} - 296 x^{9} + 519 x^{8} + 437 x^{7} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1205)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{11} q^{3} + \beta_{2} q^{4} + ( - \beta_{10} - \beta_{8}) q^{6} + (\beta_{11} - \beta_{10} + \beta_{9} + 1) q^{7} - \beta_{3} q^{8} + (\beta_{13} + \beta_{12} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{11} q^{3} + \beta_{2} q^{4} + ( - \beta_{10} - \beta_{8}) q^{6} + (\beta_{11} - \beta_{10} + \beta_{9} + 1) q^{7} - \beta_{3} q^{8} + (\beta_{13} + \beta_{12} + \cdots - \beta_1) q^{9}+ \cdots + ( - 4 \beta_{14} + \beta_{13} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 2 q^{2} + 7 q^{3} + 6 q^{4} - 5 q^{6} + 3 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 2 q^{2} + 7 q^{3} + 6 q^{4} - 5 q^{6} + 3 q^{7} - 3 q^{8} + 6 q^{9} - 10 q^{11} + 6 q^{12} + 8 q^{13} - 5 q^{14} - 16 q^{16} + q^{17} + 3 q^{18} - 30 q^{19} - 11 q^{21} + 5 q^{22} - 19 q^{23} - 14 q^{24} - 18 q^{26} + 22 q^{27} + 20 q^{28} - 12 q^{29} - 22 q^{31} + 2 q^{32} - 4 q^{33} - 29 q^{34} - 7 q^{36} + 12 q^{37} + 18 q^{38} - 17 q^{39} - 13 q^{41} + q^{42} + 25 q^{43} - 20 q^{44} - 7 q^{46} - 16 q^{47} + 22 q^{48} - 24 q^{49} - 27 q^{51} + 15 q^{52} + 4 q^{53} - 43 q^{54} - 3 q^{56} - 22 q^{57} + 20 q^{58} - 50 q^{59} - 41 q^{61} - 12 q^{62} - 6 q^{63} - 53 q^{64} + 5 q^{66} + 43 q^{67} - 5 q^{68} - 50 q^{69} - 14 q^{71} - 32 q^{72} + 10 q^{73} - 26 q^{74} - 13 q^{76} + 7 q^{77} - 3 q^{78} - 44 q^{79} + 7 q^{81} + 19 q^{82} - 7 q^{83} - 42 q^{84} + 7 q^{86} - 10 q^{87} + 28 q^{88} + 4 q^{89} - 50 q^{91} - 25 q^{92} - 22 q^{93} - 14 q^{94} + 14 q^{96} - 9 q^{97} - 2 q^{98} - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2 x^{14} - 16 x^{13} + 31 x^{12} + 99 x^{11} - 184 x^{10} - 296 x^{9} + 519 x^{8} + 437 x^{7} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( - \nu^{14} + \nu^{13} + 16 \nu^{12} - 14 \nu^{11} - 98 \nu^{10} + 73 \nu^{9} + 284 \nu^{8} - 172 \nu^{7} + \cdots - 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( - \nu^{14} + \nu^{13} + 16 \nu^{12} - 15 \nu^{11} - 97 \nu^{10} + 86 \nu^{9} + 274 \nu^{8} - 233 \nu^{7} + \cdots + 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{13} - \nu^{12} - 17 \nu^{11} + 14 \nu^{10} + 113 \nu^{9} - 71 \nu^{8} - 367 \nu^{7} + 153 \nu^{6} + \cdots + 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{14} - \nu^{13} - 17 \nu^{12} + 14 \nu^{11} + 113 \nu^{10} - 71 \nu^{9} - 367 \nu^{8} + 153 \nu^{7} + \cdots - 1 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( - \nu^{14} + 2 \nu^{13} + 16 \nu^{12} - 30 \nu^{11} - 99 \nu^{10} + 170 \nu^{9} + 293 \nu^{8} + \cdots + 1 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( - \nu^{14} + 2 \nu^{13} + 15 \nu^{12} - 31 \nu^{11} - 84 \nu^{10} + 186 \nu^{9} + 212 \nu^{8} + \cdots + 6 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( \nu^{14} - 2 \nu^{13} - 16 \nu^{12} + 31 \nu^{11} + 98 \nu^{10} - 183 \nu^{9} - 283 \nu^{8} + 509 \nu^{7} + \cdots - 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \nu^{14} - 2 \nu^{13} - 16 \nu^{12} + 31 \nu^{11} + 98 \nu^{10} - 183 \nu^{9} - 283 \nu^{8} + 509 \nu^{7} + \cdots - 5 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( \nu^{14} - 18 \nu^{12} - 3 \nu^{11} + 128 \nu^{10} + 40 \nu^{9} - 450 \nu^{8} - 193 \nu^{7} + \cdots + 57 \nu \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( \nu^{14} - 3 \nu^{13} - 15 \nu^{12} + 47 \nu^{11} + 84 \nu^{10} - 281 \nu^{9} - 210 \nu^{8} + 793 \nu^{7} + \cdots - 6 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( - 2 \nu^{13} + 2 \nu^{12} + 33 \nu^{11} - 29 \nu^{10} - 210 \nu^{9} + 157 \nu^{8} + 641 \nu^{7} + \cdots - 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + 5\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 6\beta_{3} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{14} + 2 \beta_{12} - 9 \beta_{11} + 8 \beta_{10} + \beta_{8} - \beta_{5} + \beta_{4} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{14} - \beta_{13} + 10 \beta_{12} - 11 \beta_{11} + 10 \beta_{10} + 9 \beta_{8} + \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 22 \beta_{14} - \beta_{13} + 21 \beta_{12} - 61 \beta_{11} + 51 \beta_{10} - \beta_{9} + 11 \beta_{8} + \cdots + 118 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 84 \beta_{14} - 11 \beta_{13} + 72 \beta_{12} - 86 \beta_{11} + 73 \beta_{10} - \beta_{9} + 61 \beta_{8} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 170 \beta_{14} - 13 \beta_{13} + 156 \beta_{12} - 374 \beta_{11} + 301 \beta_{10} - 14 \beta_{9} + \cdots + 510 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 556 \beta_{14} - 85 \beta_{13} + 457 \beta_{12} - 585 \beta_{11} + 473 \beta_{10} - 17 \beta_{9} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1144 \beta_{14} - 115 \beta_{13} + 1012 \beta_{12} - 2188 \beta_{11} + 1712 \beta_{10} - 129 \beta_{9} + \cdots + 2242 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3428 \beta_{14} - 573 \beta_{13} + 2727 \beta_{12} - 3700 \beta_{11} + 2887 \beta_{10} - 180 \beta_{9} + \cdots + 587 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7178 \beta_{14} - 864 \beta_{13} + 6135 \beta_{12} - 12481 \beta_{11} + 9542 \beta_{10} - 991 \beta_{9} + \cdots + 10003 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.34497
2.14166
1.96562
1.92974
1.09783
1.07739
0.596683
−0.0849802
−0.227272
−0.324166
−1.00750
−1.39546
−1.93741
−2.06795
−2.10915
−2.34497 3.01587 3.49887 0 −7.07210 −1.32864 −3.51479 6.09545 0
1.2 −2.14166 0.668121 2.58670 0 −1.43089 3.26618 −1.25651 −2.55361 0
1.3 −1.96562 0.551502 1.86368 0 −1.08405 −1.56703 0.267954 −2.69585 0
1.4 −1.92974 −0.860854 1.72391 0 1.66123 4.27878 0.532780 −2.25893 0
1.5 −1.09783 −1.80555 −0.794772 0 1.98218 0.928281 3.06818 0.259998 0
1.6 −1.07739 −0.702822 −0.839239 0 0.757210 −0.460803 3.05896 −2.50604 0
1.7 −0.596683 3.22200 −1.64397 0 −1.92252 −0.0914365 2.17430 7.38130 0
1.8 0.0849802 1.30288 −1.99278 0 0.110719 −0.925444 −0.339307 −1.30250 0
1.9 0.227272 −1.31252 −1.94835 0 −0.298298 −4.34237 −0.897348 −1.27730 0
1.10 0.324166 1.44456 −1.89492 0 0.468276 −1.26380 −1.26260 −0.913251 0
1.11 1.00750 2.74497 −0.984941 0 2.76556 1.78457 −3.00733 4.53484 0
1.12 1.39546 −2.59963 −0.0526942 0 −3.62768 1.49578 −2.86445 3.75808 0
1.13 1.93741 1.04448 1.75357 0 2.02359 2.21578 −0.477443 −1.90906 0
1.14 2.06795 −1.49139 2.27640 0 −3.08411 2.20594 0.571582 −0.775761 0
1.15 2.10915 1.77838 2.44853 0 3.75087 −3.19580 0.946028 0.162626 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(241\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.i 15
5.b even 2 1 1205.2.a.c ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.a.c ✓ 15 5.b even 2 1
6025.2.a.i 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

\( T_{2}^{15} + 2 T_{2}^{14} - 16 T_{2}^{13} - 31 T_{2}^{12} + 99 T_{2}^{11} + 184 T_{2}^{10} - 296 T_{2}^{9} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{15} - 7 T_{3}^{14} - T_{3}^{13} + 100 T_{3}^{12} - 133 T_{3}^{11} - 484 T_{3}^{10} + 970 T_{3}^{9} + \cdots - 191 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{15} - 7 T^{14} + \cdots - 191 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 3 T^{14} + \cdots - 241 \) Copy content Toggle raw display
$11$ \( T^{15} + 10 T^{14} + \cdots + 20041 \) Copy content Toggle raw display
$13$ \( T^{15} - 8 T^{14} + \cdots - 17551 \) Copy content Toggle raw display
$17$ \( T^{15} - T^{14} + \cdots - 574859 \) Copy content Toggle raw display
$19$ \( T^{15} + 30 T^{14} + \cdots + 77731 \) Copy content Toggle raw display
$23$ \( T^{15} + 19 T^{14} + \cdots + 250937 \) Copy content Toggle raw display
$29$ \( T^{15} + 12 T^{14} + \cdots + 116809 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 21531245891 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 3773372033 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 8686116791 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 336240191 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 176160469 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 27449089409 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 4073980277 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 416282129347 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 602438539877 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 236049251731 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 416966690521 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 1484049491 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 43476792713 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 2543711203 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 3592948942717 \) Copy content Toggle raw display
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